Analytical theory for the motion of spinning rigid bodies

The objective of this research is to develope analytical solutions for the translational and rotational motion of a spinning rigid body when it is subjected to constant body-fixed forces and constant moments. We consider two cases in this thesis: (1) when we have three body-fixed forces along all three axes and two body-fixed moments along two transverse axes, and (2) when we have three body-fixed forces and three body-fixed moments along all three axes. Among the aerospace applications of these analytical theories are the thrusting, spinning spacecraft problem and the thrusting, spinning-up (or spinning-down) spacecraft problem. We consider a single spinning rigid body with constant mass properties and constant body-fixed forces and moments. Also, we assume that two Eulerian angles are small and the spin axis is the z axis which may be either the maximum or minimum moment-of-inertia axis. The results of this theory are valid for axisymmetric, nearly-axisymmetric and under some circumstances, asymmetric rigid bodies. We use the analytical solutions to develope an asymptotic theory for the two aforementioned cases. Considering secular terms and finding asymptotic limits in the complete analytical solutions give insight into the behaviour of the motion. Identifying periodic and secular terms or determining the asymptotic limits (arising from a complete but complicated analytical theory) lead to simple closed-form practical solutions. Also, considering the geometrically limiting cases of a sphere, a thin rod, and a flat disk can drastically reduce the number of terms in the full-blown theory. The distillation of such simplified expressions can give the practical engineer a handle in understanding the essential motion—a back-of-the envelope calculation, a sanity check. We find asymptotic limits for angular velocities, Eulerian angles, velocity bias angle, angular momentum pointing error, transverse and axial velocities, and transverse and axial displacements